Regular immersions directed by algebraically elliptic cones

Let $M$ be an open Riemann surface and $A$ be the punctured cone in ${\mathbb{C}}^n\setminus \{0\}$ on a smooth projective variety $Y$ in ${\mathbb{P}}^{n-1}$. Recently, Runge approximation theorems with interpolation for holomorphic immersions $M\to {\mathbb{C}}^n$, directed by $A$, have been proved under the assumption that $A$ is an Oka manifold. We prove analogous results in the algebraic setting, for regular immersions directed by $A$ from a smooth affine curve $M$ into ${\mathbb{C}}^n$. The Oka property is naturally replaced by the stronger assumption that $A$ is algebraically elliptic, which it is if $Y$ is uniformly rational. Under this assumption, a homotopy-theoretic necessary and sufficient condition for approximation and interpolation emerges. We show that this condition is satisfied in many cases of interest.